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  \begin{module}[id=bbt-size]
  \importmodule[balanced-binary-trees]{balanced-binary-trees}
  \importmodule[\KWARCslides{dmath/en/cardinality}]{cardinality}

  \begin{frame}
  \frametitle{Size Lemma for Balanced Trees}
  \begin{itemize}
  \item
  \begin{assertion}[id=size-lemma,type=lemma]
  Let $G=\tup{V,E}$ be a \termref[cd=binary-trees]{balanced binary tree}
  of \termref[cd=graph-depth,name=vertex-depth]{depth}$n>i$, then the set
  $\defeq{\livar{V}i}{\setst{\inset{v}{V}}{\gdepth{v} = i}}$ of
  \termref[cd=graphs-intro,name=node]{nodes} at
  \termref[cd=graph-depth,name=vertex-depth]{depth} $i$ has
  \termref[cd=cardinality,name=cardinality]{cardinality} $\power2i$.
  \end{assertion}
  \item
  \begin{sproof}[id=size-lemma-pf,proofend=,for=size-lemma]{via induction over the depth $i$.}
  \begin{spfcases}{We have to consider two cases}
  \begin{spfcase}{$i=0$}
  \begin{spfstep}[display=flow]
  then $\livar{V}i=\set{\livar{v}r}$, where $\livar{v}r$ is the root, so
  $\eq{\card{\livar{V}0},\card{\set{\livar{v}r}},1,\power20}$.
  \end{spfstep}
  \end{spfcase}
  \begin{spfcase}{$i>0$}
  \begin{spfstep}[display=flow]
  then $\livar{V}{i-1}$ contains $\power2{i-1}$ vertexes
  \begin{justification}[method=byIH](IH)\end{justification}
  \end{spfstep}
  \begin{spfstep}
  By the \begin{justification}[method=byDef]definition of a binary
  tree\end{justification}, each $\inset{v}{\livar{V}{i-1}}$ is a leaf or has
  two children that are at depth $i$.
  \end{spfstep}
  \begin{spfstep}
  As $G$ is \termref[cd=balanced-binary-trees,name=balanced-binary-tree]{balanced} and $\gdepth{G}=n>i$,
  $\livar{V}{i-1}$ cannot contain
  leaves.
  \end{spfstep}
  \begin{spfstep}[type=conclusion]
  Thus $\eq{\card{\livar{V}i},{\atimes[cdot]{2,\card{\livar{V}{i-1}}}},{\atimes[cdot]{2,\power2{i-1}}},\power2i}$.
  \end{spfstep}
  \end{spfcase}
  \end{spfcases}
  \end{sproof}
  \item
  \begin{assertion}[id=fbbt,type=corollary]
  A fully balanced tree of depth $d$ has $\power2{d+1}-1$ nodes.
  \end{assertion}
  \item
  \begin{sproof}[for=fbbt,id=fbbt-pf]{}
  \begin{spfstep}
  Let $\defeq{G}{\tup{V,E}}$ be a fully balanced tree
  \end{spfstep}
  \begin{spfstep}
  Then $\card{V}=\Sumfromto{i}1d{\power2i}= \power2{d+1}-1$.
  \end{spfstep}
  \end{sproof}
  \end{itemize}
  \end{frame}
  \begin{note}
  \begin{omtext}[type=conclusion,for=binary-tree]
  This shows that balanced binary trees grow in breadth very quickly, a consequence of
  this is that they are very shallow (and this compute very fast), which is the essence of
  the next result.
  \end{omtext}
  \end{note}
  \end{module}

  %%% Local Variables:
  %%% mode: LaTeX
  %%% TeX-master: "all"
  %%% End: \end{document}
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